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Geometry with parallel lines and cyclic quadrilaterals

Source: European Mathematical Cup 2020, Problem J1

December 22, 2020
geometrycyclic quadrilateral

Problem Statement

Let ABCABC be an acute-angled triangle. Let DD and EE be the midpoints of sides AB‾\overline{AB} and AC‾\overline{AC} respectively. Let FF be the point such that DD is the midpoint of EF‾\overline{EF}. Let Γ\Gamma be the circumcircle of triangle FDBFDB. Let GG be a point on the segment CD‾\overline{CD} such that the midpoint of BG‾\overline{BG} lies on Γ\Gamma. Let HH be the second intersection of Γ\Gamma and FCFC. Show that the quadrilateral BHGCBHGC is cyclic. \\ \\ Proposed by Art Waeterschoot.
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